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Theorem elgiso 28497
Description: Membership in the set of group isomorphisms from  G to  H. (Contributed by Paul Chapman, 25-Feb-2008.)
Assertion
Ref Expression
elgiso  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G  GrpOpIso  H )  <->  ( F  e.  ( G GrpOpHom  H )  /\  F : ran  G -1-1-onto-> ran  H ) ) )

Proof of Theorem elgiso
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6282 . . . . 5  |-  ( g  =  G  ->  (
g GrpOpHom  h )  =  ( G GrpOpHom  h ) )
2 rneq 5219 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
3 f1oeq2 5799 . . . . . 6  |-  ( ran  g  =  ran  G  ->  ( f : ran  g
-1-1-onto-> ran  h  <->  f : ran  G -1-1-onto-> ran  h ) )
42, 3syl 16 . . . . 5  |-  ( g  =  G  ->  (
f : ran  g -1-1-onto-> ran  h 
<->  f : ran  G -1-1-onto-> ran  h ) )
51, 4rabeqbidv 3101 . . . 4  |-  ( g  =  G  ->  { f  e.  ( g GrpOpHom  h
)  |  f : ran  g -1-1-onto-> ran  h }  =  { f  e.  ( G GrpOpHom  h )  |  f : ran  G -1-1-onto-> ran  h } )
6 oveq2 6283 . . . . 5  |-  ( h  =  H  ->  ( G GrpOpHom  h )  =  ( G GrpOpHom  H ) )
7 rneq 5219 . . . . . 6  |-  ( h  =  H  ->  ran  h  =  ran  H )
8 f1oeq3 5800 . . . . . 6  |-  ( ran  h  =  ran  H  ->  ( f : ran  G -1-1-onto-> ran  h  <->  f : ran  G -1-1-onto-> ran 
H ) )
97, 8syl 16 . . . . 5  |-  ( h  =  H  ->  (
f : ran  G -1-1-onto-> ran  h 
<->  f : ran  G -1-1-onto-> ran  H ) )
106, 9rabeqbidv 3101 . . . 4  |-  ( h  =  H  ->  { f  e.  ( G GrpOpHom  h
)  |  f : ran  G -1-1-onto-> ran  h }  =  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran  H } )
11 df-giso 25024 . . . 4  |-  GrpOpIso  =  ( g  e.  GrpOp ,  h  e.  GrpOp  |->  { f  e.  ( g GrpOpHom  h )  |  f : ran  g
-1-1-onto-> ran  h } )
12 ovex 6300 . . . . 5  |-  ( G GrpOpHom  H )  e.  _V
1312rabex 4591 . . . 4  |-  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H }  e.  _V
145, 10, 11, 13ovmpt2 6413 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G  GrpOpIso  H )  =  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran  H } )
1514eleq2d 2530 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G  GrpOpIso  H )  <->  F  e.  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H } ) )
16 f1oeq1 5798 . . 3  |-  ( f  =  F  ->  (
f : ran  G -1-1-onto-> ran  H  <-> 
F : ran  G -1-1-onto-> ran  H ) )
1716elrab 3254 . 2  |-  ( F  e.  { f  e.  ( G GrpOpHom  H )  |  f : ran  G -1-1-onto-> ran 
H }  <->  ( F  e.  ( G GrpOpHom  H )  /\  F : ran  G -1-1-onto-> ran  H ) )
1815, 17syl6bb 261 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G  GrpOpIso  H )  <->  ( F  e.  ( G GrpOpHom  H )  /\  F : ran  G -1-1-onto-> ran  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2811   ran crn 4993   -1-1-onto->wf1o 5578  (class class class)co 6275   GrpOpcgr 24850   GrpOpHom cghom 25021    GrpOpIso cgiso 25023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-giso 25024
This theorem is referenced by: (None)
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